The magnitude of the drift velocity of free electrons in a conductor,…

The correct answer is Option (2) → \( \displaystyle\frac{eE\tau}{m} \)

In an electric field each free electron is accelerated between collisions, and after a collision it starts again in a random direction. Averaged over all the electrons, the initial velocities add up to zero. The average time since the last collision is the relaxation time \( \tau \). The resulting average velocity is the drift velocity.

The force on an electron is \( eE \), so its acceleration has magnitude:

    \( \displaystyle a=\frac{eE}{m} \)

Acting for the average time \( \tau \), this gives the magnitude of the drift velocity:

    \( \begin{aligned} |v_d| &= a\tau \\ &= \frac{eE\tau}{m} \end{aligned} \)

Option (1) \( \displaystyle\frac{eE}{m\tau} \) — Incorrect. Here \( \tau \) is in the denominator. A longer time between collisions would then give a smaller drift velocity. But the electron gains speed for a longer time when \( \tau \) is larger.

Option (2) \( \displaystyle\frac{eE\tau}{m} \) — Correct. The acceleration \( eE/m \) acts for the average time \( \tau \), so the drift speed is \( (eE/m)\tau \).

Option (3) \( \displaystyle\frac{eE}{m} \) — Incorrect. This is the acceleration of the electron. Its unit is \( \mathrm{m\,s^{-2}} \). A velocity needs one more factor of time, the relaxation time \( \tau \).

Option (4) \( \displaystyle\frac{E\tau}{m} \) — Incorrect. This leaves out the charge \( e \). The force on the electron is \( eE \), so \( e \) must appear in the drift velocity.

NCERT (2026-27):
Drift of Electrons and the Origin of Resistivity – “… The average value of t_i then is τ (known as relaxation time). Thus, averaging … over the N-electrons at any given time t gives us for the average velocity v_d

    \( \displaystyle \mathbf{v}_d\equiv(\mathbf{V}_i)_{average}=(\mathbf{v}_i)_{average}-\frac{e\mathbf{E}}{m}(t_i)_{average} \)

    \( \displaystyle =0-\frac{e\mathbf{E}}{m}\tau= \)\( \displaystyle -\frac{e\mathbf{E}}{m}\tau \)\( \qquad (3.17) \)

… This is the phenomenon of drift and the velocity v_d in Eq. (3.17) is called the drift velocity.”